An algorithm for partial Grundy number on trees

A coloring of a graph image is a partition image of image into independent sets or color classes. A vertex image is a Grundy vertex if it is adjacent to at least one vertex in each color class image for every image. A coloring is a partial Grundy coloring if every color class contains at least one Grundy vertex, and the partial Grundy number of a graph is the maximum number of colors in a partial Grundy coloring. We derive a natural upper bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound. In particular, this gives a linear-time algorithm to determine the partial Grundy number of a tree.